SPOD-based Spectral Galerkin Reduced Order Models

Spectral proper orthogonal decomposition (SPOD) is a space-time-optimal modal decomposition that extracts orthogonal, single-frequency coherent structures from data and is applicable to ergodic processes like many turbulent flows. Recently, new classes of frequency domain reduced-order models (ROMs) have been introduced by Lin et. al. (2019) and Towne (2021). They rely on a space-time Galerkin projection onto the SPOD modes, which makes use of the favorable properties of the decomposition. The proposed method is based on the orthogonal projection of the governing equations in the frequency domain onto an SPOD basis to compress the system at each discrete frequency. Thus, the resulting ROM is a system of algebraic equations that are solved at each frequency. An inverse Fourier transform of the spectral solution yields the entire time-domain solution over any finite time horizon of interest – no time integration is necessary. The model reduction process is applied to the Navier-Stokes equations under nonlinear colored forcing from the Reynolds-stresses for the example of a chaotic lid-driven cavity flow at Reynolds numbers of 15000 and higher. We also illustrate the explicit inclusion of triadic interactions via convolution with a linear time-varying system (bilinear convection) and elaborate the generalization to a fully quadratic nonlinearity.